OFFSET
0,3
FORMULA
G.f.: A(x) = Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
(1) G_n(x) = Series_Reversion( x/A(x^n) ),
(2) G_n(x) = x * A( G_n(x)^n ).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 18*x^4 + 78*x^5 + 415*x^6 + 2467*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n(x) satisfies: G_n( x/A(x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + 3*x^3 + 12*x^4 + 59*x^5 + 329*x^6 + 2035*x^7 +...
G_2(x) = x + x^3 + 4*x^5 + 22*x^7 + 144*x^9 + 1045*x^11 + 8159*x^13 +...
G_3(x) = x + x^4 + 5*x^7 + 35*x^10 + 289*x^13 + 2626*x^16 +...
G_4(x) = x + x^5 + 6*x^9 + 51*x^13 + 510*x^17 + 5597*x^21 +...
G_5(x) = x + x^6 + 7*x^11 + 70*x^16 + 823*x^21 + 10608*x^26 +...
G_6(x) = x + x^7 + 8*x^13 + 92*x^19 + 1244*x^25 + 18434*x^31 +...
G_7(x) = x + x^8 + 9*x^15 + 117*x^22 + 1789*x^29 + 29975*x^36 +...
where G_n(x) = x*A( G_n(x)^n ).
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, prod(k=1, m, serreverse(x/subst(A, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 45, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2012
STATUS
approved